Chapter 2

1. Chapter 2 Motion in One Dimension

2. Motion in One Dimension Introduction

3. Kinematics • Describes motion while ignoring the external agents that might have caused or modified the motion • For now, will consider motion in one dimension • Along a straight line • Motion represents a continual change in an object’s position. Introduction

4. Types of Motion • Translational • An example is a car traveling on a highway. • Rotational • An example is the Earth’s spin on its axis. • Vibrational • An example is the back-and-forth movement of a pendulum. Introduction

5. Particle Model • We will use the particle model. • A particle is a point-like object; has mass but infinitesimal size Introduction

6. Position • The object’s position is its location with respect to a chosen reference point. • Consider the point to be the origin of a coordinate system. • Only interested in the car’s translational motion, so model as a particle Section 2.1

7. Position-Time Graph • The position-time graph shows the motion of the particle (car). • The smooth curve is a guess as to what happened between the data points. Section 2.1

8. Data Table • The table gives the actual data collected during the motion of the object (car). • Positive is defined as being to the right. Section 2.1

9. Representations of the Motion of Car • Various representations include: • Pictorial • Graphical • Tablular • Mathematical • The goal in many problems • Using alternative representations is often an excellent strategy for understanding the situation of a given problem. • For example, compare the different representations of the motion. Section 2.1

10. Alternative Representations • Using alternative representations is often an excellent strategy for understanding a problem. • For example, the car problem used multiple representations. • Pictorial representation • Graphical representation • Tabular representation • Goal is often a mathematical representation Section 2.1

11. Displacement • Displacement is defined as the change in position during some time interval. • Represented as x x ≡xf- xi • SI units are meters (m) • x can be positive or negative • Different than distance • Distance is the length of a path followed by a particle. Section 2.1

12. Distance vs. Displacement – An Example • Assume a player moves from one end of the court to the other and back. • Distance is twice the length of the court • Distance is always positive • Displacement is zero • Δx = xf – xi = 0 since xf = xi Section 2.1

13. Vectors and Scalars • Vector quantities need both magnitude (size or numerical value) and direction to completely describe them. • Will use + and – signs to indicate vector directions in this chapter • Scalar quantities are completely described by magnitude only. Section 2.1

14. Average Velocity • The average velocity is rate at which the displacement occurs. • The x indicates motion along the x-axis. • The dimensions are length / time [L/T] • The SI units are m/s • Is also the slope of the line in the position – time graph Section 2.1

15. Average Speed • Speed is a scalar quantity. • Has the same units as velocity • Defined as total distance / total time: • The speed has no direction and is always expressed as a positive number. • Neither average velocity nor average speed gives details about the trip described. Section 2.1

16. Average Speed and Average Velocity • The average speed is not the magnitude of the average velocity. • For example, a runner ends at her starting point. • Her displacement is zero. • Therefore, her velocity is zero. • However, the distance traveled is not zero, so the speed is not zero. Section 2.1

17. Instantaneous Velocity • The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero. • The instantaneous velocity indicates what is happening at every point of time. Section 2.2

18. Instantaneous Velocity, graph • The instantaneous velocity is the slope of the line tangent to the x vs. t curve. • This would be the green line. • The light blue lines show that as t gets smaller, they approach the green line. Section 2.2

19. A Note About Slopes • The slope of a graph of physical data represents the ratio of change in the quantity represented on the vertical axis to the change in the quantity represented by the horizontal axis. • The slope has units • Unless both axes have the same units Section 2.2

20. Instantaneous Velocity, equations • The general equation for instantaneous velocity is: • The instantaneous velocity can be positive, negative, or zero. Section 2.2

21. Instantaneous Speed • The instantaneous speed is the magnitude of the instantaneous velocity. • The instantaneous speed has no direction associated with it. Section 2.2

22. Vocabulary Note • “Velocity” and “speed” will indicate instantaneous values. • Average will be used when the average velocity or average speed is indicated. Section 2.2

23. Analysis Models • Analysis models are an important technique in the solution to problems. • An analysis model is a description of: • The behavior of some physical entity, or • The interaction between the entity and the environment. • Try to identify the fundamental details of the problem and attempt to recognize which of the types of problems you have already solved could be used as a model for the new problem. Section 2.3

24. Analysis Models, cont • Based on four simplification models • Particle model • System model • Rigid object • Wave • Problem approach • Identify the analysis model that is appropriate for the problem. • The model tells you which equation to use for the mathematical representation. Section 2.3

25. Model: A Particle Under Constant Velocity • Constant velocity indicates the instantaneous velocity at any instant during a time interval is the same as the average velocity during that time interval. • vx = vx, avg • The mathematical representation of this situation is the equation. • Common practice is to let ti = 0 and the equation becomes: xf = xi + vx t (for constant vx) Section 2.3

26. Particle Under Constant Velocity, Graph • The graph represents the motion of a particle under constant velocity. • The slope of the graph is the value of the constant velocity. • The y-intercept is xi. Section 2.3

27. Model: A Particle Under Constant Speed • A particle under constant velocity moves with a constant speed along a straight line. • A particle can also move with a constant speed along a curved path. • This can be represented with a model of a particle under constant speed. • The primary equation is the same as for average speed, with the average speed replaced by the constant speed. Section 2.3

28. Average Acceleration • Acceleration is the rate of change of the velocity. • Dimensions are L/T2 • SI units are m/s² • In one dimension, positive and negative can be used to indicate direction. Section 2.4

29. Instantaneous Acceleration • The instantaneous acceleration is the limit of the average acceleration as t approaches 0. • The term acceleration will mean instantaneous acceleration. • If average acceleration is wanted, the word average will be included. Section 2.4

30. Instantaneous Acceleration – graph • The slope of the velocity-time graph is the acceleration. • The green line represents the instantaneous acceleration. • The blue line is the average acceleration. Section 2.4

31. Graphical Comparison • Given the displacement-time graph (a) • The velocity-time graph is found by measuring the slope of the position-time graph at every instant. • The acceleration-time graph is found by measuring the slope of the velocity-time graph at every instant. Section 2.4

32. Acceleration and Velocity, Directions • When an object’s velocity and acceleration are in the same direction, the object is speeding up. • When an object’s velocity and acceleration are in the opposite direction, the object is slowing down. Section 2.4

33. Acceleration and Force • The acceleration of an object is related to the total force exerted on the object. • The force is proportional to the acceleration, Fx ax . • Assume the velocity and acceleration are in the same direction. • The force is in the same direction as the velocity and the object speeds up. • Assume the velocity and acceleration are in opposite directions. • The force is in the opposite direction as the velocity and the object slows down. Section 2.4

34. Notes About Acceleration • Negative acceleration does not necessarily mean the object is slowing down. • If the acceleration and velocity are both negative, the object is speeding up. • The word deceleration has the connotation of slowing down. • This word will not be used in the text. Section 2.4

35. Motion Diagrams • A motion diagram can be formed by imagining the stroboscope photograph of a moving object. • Red arrows represent velocity. • Purple arrows represent acceleration. Section 2.5

36. Constant Velocity • Images are equally spaced. • The car is moving with constant positive velocity (shown by red arrows maintaining the same size). • Acceleration equals zero. Section 2.5

37. Acceleration and Velocity, 3 • Images become farther apart as time increases. • Velocity and acceleration are in the same direction. • Acceleration is uniform (violet arrows maintain the same length). • Velocity is increasing (red arrows are getting longer). • This shows positive acceleration and positive velocity. Section 2.5

38. Acceleration and Velocity, 4 • Images become closer together as time increases. • Acceleration and velocity are in opposite directions. • Acceleration is uniform (violet arrows maintain the same length). • Velocity is decreasing (red arrows are getting shorter). • Positive velocity and negative acceleration. Section 2.5

39. Acceleration and Velocity, final • In all the previous cases, the acceleration was constant. • Shown by the violet arrows all maintaining the same length • The diagrams represent motion of a particle under constant acceleration. • A particle under constant acceleration is another useful analysis model. Section 2.5

40. Kinematic Equations • The kinematic equations can be used with any particle under uniform acceleration. • The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration. • You may need to use two of the equations to solve one problem. • Many times there is more than one way to solve a problem. Section 2.6

41. Kinematic Equations, 1 • For constant ax, • Can determine an object’s velocity at any time t when we know its initial velocity and its acceleration • Assumes ti = 0 and tf = t • Does not give any information about displacement Section 2.6

42. Kinematic Equations, 2 • For constant acceleration, • The average velocity can be expressed as the arithmetic mean of the initial and final velocities. • This applies only in situations where the acceleration is constant. Section 2.6

43. Kinematic Equations, 3 • For constant acceleration, • This gives you the position of the particle in terms of time and velocities. • Doesn’t give you the acceleration Section 2.6

44. Kinematic Equations, 4 • For constant acceleration, • Gives final position in terms of velocity and acceleration • Doesn’t tell you about final velocity Section 2.6

45. Kinematic Equations, 5 • For constant a, • Gives final velocity in terms of acceleration and displacement • Does not give any information about the time Section 2.6

46. When a = 0 • When the acceleration is zero, • vxf = vxi = vx • xf = xi + vx t • The constant acceleration model reduces to the constant velocity model. Section 2.6

47. Kinematic Equations – summary Section 2.6

48. Graphical Look at Motion: Displacement – Time curve • The slope of the curve is the velocity. • The curved line indicates the velocity is changing. • Therefore, there is an acceleration. Section 2.6

49. Graphical Look at Motion: Velocity – Time curve • The slope gives the acceleration. • The straight line indicates a constant acceleration. Section 2.6

50. Graphical Look at Motion: Acceleration – Time curve • The zero slope indicates a constant acceleration. Section 2.6